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and numerical techniques
Vladislav Yastrebov
To cite this version:
Vladislav Yastrebov. Computational contact mechanics: geometry, detection and numerical tech-niques. Materials. École Nationale Supérieure des Mines de Paris, 2011. English. �NNT : 2011ENMP0043�. �pastel-00657305�
T
H
È
S
E
École doctorale n
O432 : Sciences des Métiers de l’Ingénieur
Doctorat ParisTech
T H È S E
pour obtenir le grade de docteur délivré par
l’École nationale supérieure des mines de Paris
Spécialité « Mécanique »
présentée et soutenue publiquement par
Vladislav A. YASTREBOV
le 25 mars 2011
Computational contact mechanics:
geometry, detection and numerical techniques
———–
Mécanique numérique du contact :
géométrie, détection et techniques de résolution
Directeurs de thèse : Georges CAILLETAUD,
Frédéric FEYEL
Jury
M. Peter WRIGGERS,Professeur, Leibniz Universität Hannover Président
M. Pierre ALART,Professeur, Université de Montpellier Rapporteur
M. Daniel NÉLIAS,Professeur, Université de Lyon, INSA-Lyon Rapporteur
M. Georges CAILLETAUD,Professeur, Centre des Matériaux, Mines ParisTech Directeur
M. Frédéric FEYEL,Professeur, Onera Directeur
M. Boris E. MELNIKOV,Professeur, Université Polytechnique de St. Pétersbourg Examinateur
M. Jean-François MOLINARI,Professeur, École Polytechnique Fédérale de Lausanne Examinateur
M. François COMTE,Docteur, Snecma Examinateur
MINES ParisTech
Author: Vladislav A. Yastrebov
Centre des Matériaux, MINES ParisTech CNRS UMR 7633, France
Title: Computational contact mechanics:
geometry, detection and numerical techniques
Content: 383 pages, 151 figures Specialization: Computational Mechanics Defense date: 25 March 2011
Defense place: Mines ParisTech, 60 bvd Saint Michel, Paris, France
Scientific advisers:
Prof. Georges Cailletaud
Centre des Matériaux, MINES ParisTech, CNRS UMR 7633, France Prof. Frédéric Feyel
ONERA, The French Aerospace Lab, France
Ph.D. defense committee:
President Prof. Dr.-Ing. Peter Wriggers
Leibniz Universität Hannover, Germany Reviewer Prof. Pierre Alart
Université de Montpellier, France Reviewer Prof. Daniel Nélias
Université de Lyon, INSA-Lyon, France Member Prof. Boris E. Melnikov
St. Petersburg State Polytechnical University, Russia Member Prof. Jean-François Molinari
École Polytechnique Fédérale de Lausanne, Switzerland Member Ph.D. François Comte
Snecma, France
CENTRE DES MATERIAUX P.M FOURT
The goal of this work is to derive a consistent framework for the treatment of contact problems within the Finite Element Method using the Node-to-Segment discretization. Three main components of the computational contact have been considered: geometry, detection and resolution techniques. For the sake of completeness, the mechanical aspects of contact as well as numerous numerical algorithms and methods have been discussed. A new mathematical formalism called “s-structures” has been employed through the entire dissertation. It results in a comprehensive coordinate-free notations and provides an elegant apparatus, available for other mechanical and physical applications. Several original ideas and extensions of standard techniques have been proposed and implemented in the finite element software ZéBuLoN (Z-set). Numerical case studies, presented in the dissertation, demonstrate the performance and robustness of the employed detection and resolution schemes.
Le but de ce travail était de fournir un cadre cohérent pour le traitement des problèmes de contact en utilisant une discrétisation de type nœud à segment. Trois aspects principaux de la mécanique numérique du contact ont été particulièrement considérés : la description de la géométrie, le problème de détection de contact et les techniques de résolution. Le manuscrit contient cependant une présentation complète de la mécanique du contact et des algorithmes numériques qui lui sont attachés. Un nouveau formalisme mathématique – les s-structures – est employé dans l’ensemble de la thèse. Il fournit un cadre de formulation intrinsèque qui permet d’exprimer de façon compacte un grand nombre de problèmes de mécanique et de physique. La thèse propose plusieurs idées originales et des extensions des techniques classiques, qui ont toutes été mises en œuvre dans le code de calcul par éléments finis ZéBuLoN (Z-set). Plusieurs études de cas, présentées dans la thèse, viennent démontrer les performances et la robustesse des méthodes numériques utilisées pour la détection et la résolution.
The three years of my PhD thesis at the Centre des Matériaux of the École des Mines de Paris were a fruitful period of my personal and professional development. The wideness of the subject and the freedom I had in its investigation allowed me to develop many aspects of the computational contact mechanics under a wise guidance of my scientific advisers Georges Cailletaud and Frédéric Feyel, always helpful and open to the new ideas. I highly appreciate their deep knowledge and top professionalism as well as the kind and respectful attitude towards me. For the possibility to work with them I am obliged to the recommendation of Boris Evgenievich Melnikov, whose permanent help and attention I greatly acknowledge.
I thank a lot my dear friends Nikolay Osipov and Djamel Missoum-Benziane for their constant support, friendship, help and the worm atmosphere they created. I am also very much obliged to Djamel for my rapid improvements in French. Further, I am grateful to Bahram Sarbandi who helped me a lot to set up in France and to Yoann Guilhem, Julian Frachon and Guillaume Abrivard for their open-mindedness and the “accueil chaleureux en France”. I was glad to spend the 3 years of my thesis in the same office with my colleague and friend, Sophie Cartel.
I would like to thank Lingtao Sun, Laurent Maze, Florine Maes, Guillaume Martin from the new generations of PhD students for their friendliness. I deeply appreciated the collaboration and friendly interaction with Julian Durand and Michael Fischlschweiger. I thank my colleagues Ozgur Aslan, Guruprasad Padubidri, Saber El-Arem, Jarmila Savkova, Prajwal Sabnis, Johann Rannou, Jean-Didier Garaud, Vincent Chiaruttini, Ibrahima Gueye, Eva Héripré, Henry Proudhon, Matthieu Maziere and many others for their kind attitude. I am also grateful to my elder colleagues and advisers André Pineau, Jean-Louis Chaboche, Esteban Busso, David Ryckelynck, Samuel Forest, Stephane Quilici, Farida Azzous, Françoise di Rienzo, Anne-Francoise Gourgues and many others for their personal example, support and encouragements.
A special thanks is reserved to Liliane Locicero, Konaly Sar, Odile Adam, Isabelle Olzenski and Anne Piant for their permanent help and moreover, for their empathy and friendly attention. I thank my french teacher Cécile Brossaud for interesting lessons as well as for her patience and creativity.
I thank Siarhei Dubouski and Konstantin Kuzmenkov for bringing the “Russian spirit” to the laboratory’s life. Also I would like to thank Olga Trubienko, now my friend, for our fruitful collaboration on some aspects of my thesis.
of science: Vladimir Alexandrovich Palmov, Boris Alexandrovich Smolnikov, Victor Alexeevich Pupyrev, Pavel Andreevich Zhilin, Victor Nilovich Naumov, Anton Miroslavovich Krivtsov, Artem Semenovich Semenov and Sergey Nikolaevich Kolgatin. I thank my school teacher, Mikhail Sergeevich Zhitomirsky, who gave me a solid base in Mathematics.
I appreciated that Peter Wriggers, Boris E. Melnikov, Jean-François Molinari and François Comte found the time to participate in the committee of my PhD defense. I am very grateful to Pierre Alart and Daniel Nélias for their attentive review of my thesis.
The financial support of the CNRS-SNECMA (grant #47900) is greatly acknowledged as well as the efficient collaboration with François Comte.
Very sincere thanks are addressed to my dear parents, who gave me the life, love and everything they could to make my life happy. I greet my dear brother Igor and thank him for his friendship, attention and brotherhood. And I thank my beloved wife Alexandra for her love, patience, support in my decisions, wise advices and for giving birth to our son Andrey, whose smile makes every day of my life sunny.
Vectors and tensors:
- Scalar (zero-order tensor) – small latin and greek letters: a, α, b, . . .
- Vector (first-order tensor) – underlined small bold latin and greek letters:
c, β, d, . . .
- Second-order tensor – capital bold latin letters underlined twice:
E
=, F=, . . .
- Higher order tensor – capital bold latin letters underlined twice with upper left index of order:
3
G=,4H=, . . .
V-Vectors and V-tensors:
- V-scalar (“vector of scalars”) – small latin and greek letters underlined by a wave: i ∼, γ∼, · · · ∈ m 1S n 0
- V-vector (“vector of vectors”) – small latin and greek letters underlined by a line and a wave:
j ∼, ε∼, · · · ∈ m 1S n 1
- V-tensor (“vector of tensors”) – capital bold latin letters underlined by a double line and a wave:
K= ∼, L=∼, · · · ∈ m 1S n 2 T-Vectors and T-tensors:
- T-scalar (“tensor of scalars”) – capital bold latin letter underlined by a double wave: M ≈, N≈, · · · ∈ m 2S n 0
- T-vector (“tensor of vectors”) – small latin and greek letters underlined by a line and a double wave:
o ≈, η≈, · · · ∈ m 2S n 1
P = ≈ , Q = ≈ , · · · ∈m2S n 2
Vector and tensor operations:
- a – euclidean norm of a vector; - det A= – determinant of a tensor; - I= – unit tensor;
- I
≈– unit t-scalar; - trA= – trace of a tensor; - A=−1– inverse of tensor; - A=T – transpose of tensor; - i A = · j B == i+j−2 C
= – scalar or dot product; - i A = × j B == i+j−1 C
= – vector or cross product; - i A = ⊗ j B == i
A=jB==i+jC= – tensor product;
- i A = · · j B == i+j−4 C = – tensor contraction. Other operations: - (•)·= d•
dt – full time derivative; - δ(•), ∆(•) – first variations; - ¯δ(•), ¯∆(•) – full first variations; - ∆δ(•) – second variation; - ¯∆ ¯δ(•) – full second variation; - ∇ ⊗ (•) – gradient;
- ∇ · (•) – divergence; - ∇ × (•) – rotor.
- hxi = 1
2(x + |x|) – Macaulay brackets;.
- [•, •]; (•, •); (•, •] – closed, open, open-closed intervals;
- ∀, ∃, ∃!, ∃!!, ∄ – for all, exists, exists only one, exists infinitely many, does not exist;
- ⇒, ⇐, ⇔ – sufficient, necessary, sufficient and necessary conditions; - min, max, ext, sup, inf – minimum, maximum, extremum, supremum,
infimum;
- gmin, gmax – global minimum, global maximum; - i = 1, n – i changes from 1 to n.
Abbreviations:
- PM, LMM, ALM – penalty, Lagrange multiplier, augmented Lagrangian methods;
- FEM, FEA – Finite Element Method, Finite Element Analysis; - CAD – Computer-Aided Design;
- NTN, NTS – Node-to-Node, Node-to-Segment discretizations; - MPC – Multi-Point Constraints;
- PDN – Partial Dirichlet-Neumann;
- SDMR, MDMR – Single Detection Multiple Resolution, Multiple Detection Multiple Resolution.
~
Remark on Macaulay brackets, dist(., .) and θ(.) functions.
Throughout the dissertation we use the notation of Macaulay brackets hxi =( x0, x < 0, x ≥ 0, ; h−xi =( −x, x ≤ 0,0, x> 0
Theθ function is a similar notation widely used in both engineering and mathematical literature θ(x) = max(x, 0) =( x, x ≥ 0, 0, x < 0 ; θ(−x) = min(x, 0) = ( −x, x ≤ 0,0, x> 0
dist(x, Ω) = 0,
x ∈ Ω,
where dist(x, ∂Ω) is a somehow defined distance from point x to the closure of the set Ω. For example, in the simplest case Ω = R−, x ∈ R, then ∂R− =0
dist(x, R−) = ( x0, x < 0, x ≥ 0,; dist(x, R+) = ( −x, x ≤ 0,0, x> 0.
All these functions are equivalent for the considered case and interchangeable, so the reader is invited to interpret the Macaulay brackets as one of above mentioned functions to which he is more accustomed.
1 Introduction to contact mechanics 5
1.1 Historical remark . . . 8
1.1.1 The augmented Lagrangian method . . . 10
1.2 Basics of the numerical treatment . . . 12
1.2.1 Contact detection . . . 13
1.2.2 Contact discretization . . . 14
1.2.3 Contact resolution . . . 17
1.3 Motivation and overview . . . 18
2 Geometry in contact mechanics 21 2.1 Introduction . . . 22
2.2 Interaction between contacting surfaces . . . 27
2.2.1 Normal gap . . . 28
2.2.2 The closest point . . . 33
2.2.3 Aircraft’s shadow projection method. . . 39
2.2.4 Tangential relative sliding . . . 42
2.2.5 First order variations . . . 45
2.2.6 Second order variations . . . 52
2.3 Numerical validation . . . 61
2.4 Discretized geometry . . . 65
2.4.1 Shape functions and finite elements . . . 65
2.4.2 Geometry of contact elements . . . 68
2.5 Enrichment of contact geometry. . . 74
2.5.1 Example of enrichment . . . 87 3 Contact detection 91 3.1 Introduction . . . 92 3.2 All-to-all detection . . . 96 3.2.1 Preliminary phase . . . 96 3.2.2 Detection phase . . . 100
3.3 Bucket sort detection . . . 106
3.3.1 Preliminary phase . . . 106
3.3.2 Numerical tests . . . 107
3.3.3 Detection phase . . . 110
3.3.4 Multi-face contact elements . . . 111
3.4 Validation and performance . . . 113
3.5 Case of unknown master-slave discretizations . . . 115
3.6 Parallel contact detection . . . 120
3.6.1 General presentation . . . 120
3.6.2 Single Detection, Multiple Resolution approach . . . 121
3.6.3 Multiple Detection, Multiple Resolution approach . . . . 122
3.6.4 Scalability test . . . 124
3.7 Conclusion . . . 126
4 Formulation of contact problems and resolution methods 127 4.1 Unilateral contact with a rigid plane . . . 128
4.1.1 Interpretation of contact conditions . . . 131
4.1.2 Friction. . . 133
4.1.3 Interpretation of frictional conditions . . . 137
4.1.4 Non-classical friction and adhesion laws . . . 140
4.2 Unilateral contact with an arbitrary rigid surface . . . 146
4.2.1 Non-penetration condition . . . 147
4.2.2 Hertz-Signorini-Moreau’s contact conditions . . . 150
4.2.3 Interpretation of contact conditions . . . 151
4.2.4 Frictional conditions and their interpretation . . . 152
4.3 Contact between deformable solids . . . 153
4.3.1 General formulation and variational inequality. . . 153
4.3.2 Remarks on Coulomb’s frictional law . . . 159
4.4 Variational equality and resolution methods . . . 162
4.5 Penalty method . . . 162
4.5.1 Frictionless case. . . 163
4.5.2 Example . . . 165
4.5.3 Nonlinear penalty functions. . . 168
4.5.4 Frictional case . . . 170
4.6 Method of Lagrange multipliers. . . 173
4.6.1 Frictionless case. . . 174
4.6.2 Frictional case . . . 175
4.6.3 Example . . . 178
4.7 Augmented Lagrangian Method . . . 185
4.7.1 Introduction . . . 185
4.7.2 Application to contact problems . . . 188
4.7.3 Example . . . 195
5 Numerical procedures 203 5.1 Newton’s method . . . 204
5.1.1 One-dimensional Newton’s method . . . 204
5.1.2 Multidimensional Newton’s method. . . 207
5.1.3 Application to nondifferentiable functions . . . 208
5.1.4 Subdifferentials and subgradients . . . 209
5.1.5 Generalized Newton method . . . 213
5.2 Return mapping algorithm. . . 217
5.3.1 Introduction . . . 224
5.3.2 Contact elements . . . 228
5.3.3 Discretization of the contact interface . . . 231
5.3.4 Virtual work for discretized contact interface . . . 233
5.3.5 Linearization of equations . . . 235
5.3.6 Example . . . 237
5.4 Residual vectors and tangent matrices . . . 238
5.4.1 Penalty method: frictionless case . . . 238
5.4.2 Penalty method: frictional case . . . 240
5.4.3 Augmented Lagrangian method: frictionless case . . . . 248
5.4.4 Augmented Lagrangian method: frictional case . . . 250
5.5 Method of partial Dirichlet-Neumann boundary conditions. . . 257
5.5.1 Description of the numerical technique . . . 257
5.5.2 Frictionless case . . . 258
5.5.3 Frictional case . . . 263
5.5.4 Remarks . . . 263
5.6 Technical details . . . 265
5.6.1 Rigid master surface . . . 265
5.6.2 Multi-face contact elements and smoothing techniques . 266
5.6.3 Heterogeneous friction . . . 268
5.6.4 Short remarks . . . 270
6 Numerical examples 273
6.1 Two dimensional problems . . . 274
6.1.1 Indentation by a rigid flat punch . . . 274
6.1.2 Elastic disk embedded in an elastic bored plane . . . 280
6.1.3 Indentation of an elastic rectangle by a circular indenter 284
6.1.4 Axisymmetric deep cup drawing . . . 286
6.1.5 Shallow ironing . . . 290
6.1.6 Axisymmetric post buckling of a thin-walled cylinder. . 295
6.2 Three dimensional problems. . . 299
6.2.1 Accordion post buckling folding of a thin-walled tube . 299
6.2.2 Hydrostatic extrusion of a square plate through a circular hole . . . 303
6.2.3 Frictional sliding of a cube on a rigid plane . . . 310
7 Conclusions and perspectives 315
7.1 Original contributions . . . 316
7.1.1 Geometry . . . 316
7.1.2 Detection. . . 317
7.1.3 Resolution . . . 318
7.2 Intermediate results and perspectives . . . 319
7.2.1 Normal contact of rough surface . . . 319
A Vectors, tensors and s-structures 325
A.1 Fundamentals . . . 325
A.2 Vector space basis . . . 330
A.2.1 Transformation matrices, covariant and contravariant objects . . . 332
A.2.2 Gradient operator or Hamilton’s operator . . . 334
A.3 Sub-basis, vector function of v-scalar argument . . . 336
A.4 Tensors . . . 339
A.5 Tensor as a linear operator on vector space . . . 346
A.6 S-structures . . . 347
A.6.1 Formal definition, notations and types . . . 350
A.6.2 Simple operations . . . 353
A.6.3 Invariant s-structures. . . 354
A.6.4 Scalar products of V-vectors . . . 356
A.6.5 Inverse v-vector. . . 360
A.6.6 Isomorphism of s-space and tensor space . . . 361
A.6.7 Tensor product of v-vectors . . . 364
Introduction to contact
mechanics
Résumé de Chapitre
1
«Introduction à la mécanique de
contact»
Ce chapitre présente une brève histoire des développements de la mécanique du contact, de sa naissance jusqu’à nos jours. On insiste en particulier sur les aspects numériques et on présente une formulation mathématique rigoureuse des problèmes associés. La littérature concernant la méthode du Lagrangien augmenté est exposée en détail.
De plus, ce chapitre introduit les notions de base qu’on exploite au cours du manuscrit. Pour donner une vue globale sur le traitement numérique des problèmes de contact, on éclaircit toutes les étapes de l’algorithme : la détection du contact, la discrétisation et la résolution. On présente la revue des possibilités existantes pour chaque de ces étapes et on argumente les choix qui seront effectués par la suite : méthode de «bucket sort» modifiée pour la détection, méthode du Lagrangien augmenté et méthode de pénalisation pour la résolution et la discrétisation «Nœud-à-Segment». On expose également les éléments qui ont motivé ce travail et on présente le plan du manuscrit.
From a mechanical point of view, at macroscale, contact is a notion for all types of interactions between separate bodies coming in touch. Direct contact between solids allows to transfer a load, a heat and an electric charge from one body to another. The physics of the contact interaction is particularly rich and complicated, due to the multiscale and multiphysical nature of the phenomenon. The branch of mechanical engineering studying this interaction is called tribology – a science of relative motion of interacting surfaces in a comprehensive framework combining mechanical, physical and chemical effects at different scales. This dissertation presents the mathematical description and modeling of the mechanical aspects of this interaction.
Contact problems in mechanics of deformable solids can be singled out in a particular class. There are several reasons for that. Contact occurs in
boundary conditions imposed on both contacting surfaces. At the same time, the contact interface itself cannot be simply considered as an internal surface. In an idealized case, the contact interface is a zero thickness layer, which sustains only compressive stress in the direction orthogonal to the contact interface (Fig.1.1,a.), any stretching leads to vanishing of the contact interface (Fig.1.1,b.). In case of frictionless contact, the contact interface contrary to an ordinary internal surface, does not sustain any tangential efforts, which allows two surfaces slide relatively to each other (Fig. 1.1,a.). In case of frictional contact, tangential resistance of the contact interface is similar to the resistance of an elasto-plastic material with saturation. For example, in case of the classic Coulomb’s friction law in stick state, the contact interface represents an internal surface – no separation, no tangential sliding – locally both surfaces remain glued to each other (Fig.1.1,c.). If a critical shear stress is reached, the surfaces start to slip relatively to each other, however the nonzero shear stress remains equilibrated (Fig. 1.1,d.). It follows from this simple representation that the contribution of the contact interface to the energy of the system is always zero except in the case of frictional slip.
Figure 1.1: Analogy between contact interface and internal interface: a – frictionless contact sustains compressive stress in the local reference frame,
b– any stretching leads to vanishing of contact interface, c – frictional contact
interface can transfer shear stress; d – in Coulomb’s friction law in stick state there is no relative sliding up to reaching a critical shear stress.
Mechanical problems are classically formulated as boundary value problems, where a governing differential equations should be fulfilled within the domain Ω and ordinary boundary conditions are imposed on the domain’s closure ∂Ω. The balance of virtual work yields a weak (integral) form of this boundary value problem, which presents a basis on which the structural Finite Element Method is constructed. Contact constraints are formulated as sets of inequalities. Such a formulation is not usual for boundary value problems. The rigorous construction of a variational principle leads to a variational inequality
instead of a classic variational equality. Such a new mathematical structure requires new solution approaches. The problem becomes even more complex when a frictional effect is assumed at the interface. Coulomb’s friction law states that tangential resistance depends upon the normal contact pressure, but the latter is known only if the solution is known. Roughly speaking, the boundary conditions are solution dependent, which naturally leads to difficulties in the formulation of the frictional contact problem. Moreover, the nature of Coulomb’s friction law yields a nonsmooth energy functional resulting in even more difficulties from a numerical point of view. As pointed out in the book of Kikuchi and Oden [Kikuchi 88] “Frictional contact problem between continuous deformable solids involves formidable mathematical difficulties”.
Another mathematical difficulty in contact mechanics arises from a rigorous description of continuous interacting surfaces. First, contacting bodies may penetrate each other or be separated. In both cases, a bijection between points of the contacting surfaces does not always exist. Second, the finite element discretization results only in piecewise smooth contacting surfaces, which leads to mathematical and numerical difficulties. Third, a considerable effort has to be undertaken to derive a rigorous linearization of the variational principle, which in turn requires second order variations of the normal gap and the tangential sliding, which is not an easy task. Basic knowledges of differential geometry is needed to obtain the relevant quantities.
The discretization of the contact interface is a third challenge in computational contact mechanics. A simple and stable discretization for conforming meshes, i.e. each node on one contacting surface has a corresponding node on the other surface, can be established only in case of small deformations and infinitely small relative sliding. Such a discretization is called Node-to-Node. A less simple but multipurpose discretization implies the creation of contact pairs consisting of a node of one surface and a corresponding segment of the other surface. This approach is known as Node-to-Segment discretization. However, this discretizations does not fulfill the so called Babuška-Brezzi conditions and leads to an unstable discretization. Recently, new techniques based on segment-to-segment discretizations – Nitsche and mortar methods – have been successfully introduced in computational contact mechanics, however, the computer implementation of these methods for a general case presents a real challenge both from algorithmic and technical points of view. Seeking for a stable and relatively simple discretization of the contact interface is still in progress.
All forementioned difficulties are related to the resolution phase of the contact algorithm. It follows the detection phase, which determines the contacting pairs on discretized surfaces. At first glance, the detection phase is a standalone task, but in reality it appears to be strongly connected with the discretization type of the contact interface, with the definition of the gap function and with the type of contact (e.g., simple contact or self-contact). The detection phase may present a bottleneck for an efficient treatment of contact problems, both for rapidity and robustness. The contact detection becomes one of the most crucial points for an efficient parallelization of the whole
resolution scheme. Elaboration and implementation of an efficient contact detection algorithm is an absolute necessity for a robust and fast Finite Element Analysis of large contact problems.
In this introduction, after a brief historical review, the main notions of contact mechanics and related applications will be given, followed by a short presentation of the general concept of contact treatment in the framework of the Finite Element Method and implicit integration. The questions of detection, discretization and resolution will then be addressed in first approximation. Further, the main physical aspects of frictional contact will be introduced. Finally the contents of the dissertation will be presented.
1.1
Historical remark
The modern contact mechanics is about 130 years old. It started in 1882 with the publication of Hertz’s famous paper On the contact of elastic solids [Hertz 82], which gives the solution for frictionless contact between two ellipsoidal bodies. This problem had arisen from the problem of the optical interference between glass lenses. Futher developments in the contact theory appear only in the beginning of XXth century in application to railways, to reduction gears and to rolling contact bearing industry. Progress in contact mechanics was associated with removing the restrictions of the Hertz theory, such as pure elasticity, frictionless and small deformations. A large contribution has been made by the Russian school of mechanicians, starting from Galin [Galin 53], [Galin 76] and Muskhelishvili [Muskhelishvili 66]. A synthesis of analytical solutions and approaches for contact problems can be found in monographs [Lurie 70], [Alexandrov 83], [Johnson 94], [Goryacheva 98], [Goryacheva 01], [Vorovich 01], etc.
Since the analytical solution is achievable only for a few simple geometries, boundary conditions, and mostly for linear materials, only rough approximations based on these solutions can be established for complicated frictional contact problems. These problems come from industrial needs and are usually coupled with complex geometries, boundary conditions and non-linear materials. For that reason, with approaching computer age, more and more numerically based semi-analytical solutions for contact problems appear. But it is still not sufficient to answer the industrial demand for a fast and accurate resolution of contact problems, which may include friction, wear, adhesion, large deformations, large sliding and non-linear material.
Since 1965 (NASTRAN) the Finite Element Method (FEM) becomes one of the most usable and efficient tools for the treatment of problems in structural mechanics. In order to fulfill industrial demands related to contact problems, the scientific society worked out a rigorous mathematical framework valid for incorporation of the contact in the Finite Element Method. This task required formidable efforts from the mathematico-mechanical community. First, the frictionless Signorini’s problem (unilateral contact between a deformable body and a rigid foundation) has been treated, further the developed approaches have been extended to the case of unilateral frictional contact in small and
large deformations and finally to bilateral1or multibody contact. At the same time, the engineering practice tested the solution schemes and proposed new challenging tasks. The work on a stable approach for treatment of large sliding frictional contact is still in progress.
The history of the computational contact began in 1933 with the works of Signorini who was the first who formulated the general problem of the equilibrium of a linearly elastic body in frictionless contact with a rigid foundation [Signorini 33], [Signorini 59]. The works of Fichera represents the first treatment of questions of existence and uniqueness of the variational inequalitiesarising from the minimization of functionals on convex subsets of Banach spaces, which yields from his rigorous analysis of a class of Signorini’s problems [Fichera 63], [Fichera 64], [Fichera 72]. Variational inequality is a new structure in the field of the optimization theory; new approaches are required to make use of such formulations for practical problems of physics and mechanics. “Inequalities in mechanics and physics” by Duvaut and Lions (first published in French and rapidly translated in English [Duvaut 76]) was a real scientific breakthrough in this direction, the authors investigated the solution of frictional contact problems and large deformation contact. Among the early relevant contributions related to contact problems, the following can be enumerated Cocu [Cocu 84], Panagiotopoulos [Panagiotopoulos 85], Rabier et al[Rabier 86]. A consistent description of the variational inequality approach to contact problems is given in the book by Kikuchi and Oden [Kikuchi 88], where among other important results the existence and uniqueness of the solution of Signorini’s problem is proven. Stability questions of contact problem solution have been discussed by Klarbring [Klarbring 88]; examples of non-uniqueness or non-existence were demonstrated by Klarbring [Klarbring 90] and Martins et al. [Martins 94]. The existence and uniqueness results for dynamic contact problems can be found in Martins and Oden [Martins 87], Jarusek and Eck [Jarusek 99] and others.
The frictionless contact problem formulated as a variational inequality presents a special type of minimization problems with inequality constraints, which can be efficiently treated in a standard manner (penalty method, Lagrange multiplier method, augmented Lagrangian method, etc.). Unfortunately, there is no associated minimization principle for the frictional contact problem [Kikuchi 88], [Mijar 00]. Such a problem is rather complicated and unusual for optimization theory since the energy of the system (objective function) depends on the frictional status which depends on the normal contact pressure, which in turn depends on the displacements, i.e. on the solution of the problem which again depends on the energy of the system. Since there is no smooth energy functional associated with the frictional contact problem, its formulation and resolution present real challenges.
The assumption of a known a priori contact interface on the current computational step results in a reformulation of the variational inequality into a variational equality problem with a special contact term; the form of this term
1bilateral contact between two or more deformable solids, in contrast to unilateral contact
depends on the method chosen to enforce the contact constraints. Among the well-known and widely used methods there are: barrier and penalty methods, Lagrange multiplier methods and their combinations. Another branch of methods makes use of different techniques from mathematical programming: application of the simplex method to contact problems can be found in [Chand 76], parametric quadratic programming method is employed in [Klarbring 86], [Zhong 88]. Separately from these two branches, there is a group of direct methods, which treats the contact problem independently from the structural one: the flexibility method proposed by Francavilla and
Zienkiewicz[Francavilla 75], modified and improved by Jean [Jean 95], rarely
mentioned in the scientific literature, in practice this method demonstrates a higher robustness and rapidity in comparison to ordinary methods if the number of nodes in contact remains moderate. But this method is not applicable for large contact problems and its parallelization is hardly possible. A detailed description of the method and its application can be found in [Wronski 94]. A complete list of methods used for the numerical treatment of contact problems can be found in [Wriggers 06] and [Laursen 02].
1.1.1 The augmented Lagrangian method
As mentioned in the previous section, the assumption of a known a priori contact surface allows to replace the variational inequality by a variational equality with an additional contact term. The form of this contact term depends upon the choice of the optimization method; the most usable in contact mechanics are the Lagrange multiplier method, the linear penalty method and an augmented Lagrangian method, the two latter methods are implemented in leading modern finite element analysis softwares: ANSYS [Bhashyam 02], [Oatis 07], [ANS 05], ABAQUS [ABA 07], COMSOL [COM 10] and others. In this dissertation all forementioned methods are considered, but a particular attention is paid to the augmented Lagrangian method, possessing several advantages in comparison to other methods.
Within the framework of classical Lagrange multiplier method (LMM), contact conditions are exactly satisfied by the introduction of extra degrees of freedom called Lagrange multipliers. The constrained minimization problem converts into an unconstrained saddle point problem often called min-max problem. Due to inequality constraints this formulation has to be considered in combination with an active set strategy [Luenberger 03], [Murty 88], i.e. a check and update of active and passive constraints should be integrated in the convergence loop. Moreover, the additional degrees of freedom of the LMM introduce supplementary computational efforts. Penalty method (PM) is simple to implement and to interpret from the physical point of view, but, on the other hand, the contact conditions are fulfilled exactly only in case of the infinite penalty parameter which results in ill-conditioning of the numerical problem. The augmented Lagrangian method (ALM) is a sort of Lagrange multiplier formulation regularized by penalty functions. It yields a smooth energy functional and fully unconstrained problem, resulting in exact fulfillment of contact constraints with a finite value of the penalty parameter.
In this section a few historical remarks concerning the augmented Lagrangian method are given. For a more detailed background the reader is referred to the articles and books cited below.
The augmented Lagrangian method has been proposed in the first raw approximation by Arrow and Solow in 1958 [Arrow 58b]. Further a more elaborated version of the ALM method for optimization problems subjected to equality constraints has been independently proposed by
Hestenes [Hestenes 69] and Powell [Powell 69] in 1969. As mentioned by Pietrzak[Pietrzak 97] it was proposed "rather in an intuitive way" and a lot
of questions have not been considered. The way to apply the ALM method to optimization problems with inequality constraints has been developed by
Rockafellar[Rockafellar 70], [Rockafellar 73b] and Wierzbicki [Wierzbicki 71].
Using the augmented Lagrangian method as well as the Lagrange multiplier method leads to the saddle point problem, i.e. the objective function is to be minimized by "ordinary" primal variables (e.g., displacement degrees of freedom (dof) in the displacement based FEM) and is to be maximized by dual variables - Lagrange multipliers which represent contact stresses. All forementioned authors approach this min-max (saddle point) problem by an independent consecutive updating of the primal and dual degrees of freedom. An algebraic formula is used to update the Lagrange multipliers at each iteration step and consequently a standard minimization procedure is used to update the primal degrees of freedom. This idea has been worked out by Powell [Powell 69]. Nowadays such an approach is employed under the name of Uzawa’s algorithm and the full method is referred as a nested augmented Lagrangian algorithm. Another approach has been developed by
Fletcher[Fletcher 70]. It consists in a continuous minimization of the resulting
saddle problem with a simultaneous update of both primal and dual variables. One of the first applications of the augmented Lagrangian method to frictionless contact problem can be found in Glowinski and Le
Tallec[Glowinski 89] and Wriggers, Simo and Taylor [Middleton 85]. The first
application of the augmented Lagrangian method with Uzawa’s algorithm to frictional problems has been reported by Simo and Laursen [Simo 92]. The first successful attempt to apply the coupled augmented Lagrangian method to frictional contact problems has been undertaken by Alart [Alart 88], and
Alart and Curnier[Alart 91]. The augmented Lagrangian approach has been
elaborated by developing the perturbation approach to convex minimization as proposed in [Rockafellar 70] and first applied by Fortin [Fortin 76] to visco-plastic flow problems (rather similar to frictional contact problems).
Further developments of the ALM method to large deformations, large sliding and nonlinear materials can be found in [Heegaard 93], [Mijar 04a], [Mijar 04b], etc. A comprehensive investigation on the implementation of the ALM method in the framework of the Finite Element Method to large deformation frictional contact problems has been carried out by Pietrzak and
Curnier [Pietrzak 97], [Pietrzak 99]. The attempts to work out a technique
for penalty parameter updating are worth mentioning, since it became a crucial factor for convergence of the ALM. A direction was proposed in early works [Hestenes 69] and [Powell 69]. The need was mentioned
by Rockafellar [Rockafellar 73b], discussed in [Alart 97] and an approach has been proposed by Mijar and Arora [Mijar 04a], [Mijar 04b]; another phenomenological approach has been proposed in [Bussetta 09]. An early attempt to parallelize the ALM has been undertaken by Barboteu and
Alart[Barboteu 99] for particular structures.
The augmented Lagrangian method combines advantages of both methods LMM and PM and avoids their drawbacks, precisely it converges to the exact solution for a finite value of the penalty coefficient and if a nested update of dual variables is used, there is almost no additional computational efforts. Following Pietrzak, we would like to emphasis the smoothing effect of the ALM which is not the only advantage over ordinary LMM. Even in case of a smooth objective function the ALM method shows its superiority. The ordinary LMM does not fully reduce the optimization problem with inequality constraints to an unconstrained problem, since the condition of positivity of the Lagrange multipliers λ ≥ 0 has to be satisfied. The ALM method does not have this restriction and therefore is better for practical use. An elaborated presentation of the method will be given in Section4.7.
1.2
Basics of the numerical treatment of contact problems
The part of the implicit Finite Element code aimed at the treatment of contact problems consists in the following steps: contact detection, construction of “contact elements”, incorporation of these elements with associated residual vectors and tangential matrices in the general nonlinear problem and finally resolution of the resulting problem. Here we give the main ideas and a general view of these steps, which will be presented in details further in the corresponding chapters.
Contact elementsare a kind of “bridge elements” between locally separated but potentially interacting surfaces. Each contact element contains components (nodes, edges, segments or their parts) of both surfaces; the composition of these components depends upon the choice of the contact discretization method. Each contact element has its own vector of unknowns, residual vector and tangential matrix, which are assembled with unknowns, residual vectors and matrices of ordinary structural elements. The set of unknowns and the structure of the residual vector and the tangential matrix are determined by the resolution method. For example, in addition to primal unknowns (e.g. displacement) contact elements may contain dual unknowns (Lagrange multipliers) representing contact stresses.
The Contact detection is a step preceding all others. The aim of this step is to create contact elements containing the proximal components of both surfaces which may contact on the current solution step. As a consequence the detection algorithm is based on a search for the closest components and presents a particular algorithmic task. The criterion of proximity is either provided by a user or is chosen automatically based on boundary conditions and/or discretization of contacting surfaces. In order to incorporate contact elements in the resolution cycle, they should be created before a contact occurs
and if needed should be removed and recreated at each solution step. Contrary to this scheme, in case of explicit integration, the searching step consists in the detection of penetration, which has already occurred.
In order to treat contact problems, from the programmer’s point of view, a standard finite element code has to be complemented by
1. a class governing contact; 2. a contact detection algorithm; 3. a class of contact elements;
4. the corresponding residual vectors and tangential matrices.
1.2.1 Contact detection
The development of numerical methods and the increasing demands on complexity (large deformation, large sliding, self-contact) and size of problems in computational contact mechanics entailed the development of contact detection techniques. As previously mentioned the contact detection presents a purely algorithmic task and is strongly connected with the discretization of the contact interface. For example, in the case of Node-to-Node discretization, the contact detection consists simply in establishing close pairs of nodes: nodes from one surface form pairs with their closest opponents from another surface. Since the Node-to-Node discretization is limited to small deformation and infinitely small slidings, once created contact pairs do not change during the solution steps. Node-to-segment discretization requires a more elaborated detection procedure: for nodes of one surface (slave) the closest point on the other surface (master) has to be found, the master segment possessing this point complemented by the slave node forms a Node-to-Segment contact element.
This simple detection procedure generates several difficulties. First, the detection of the closest point on the master segments may fail if the slave node is not sufficiently close to the master surface or if the latter is not smooth, which is the case in case of finite element discretization of the surface. The numerical scheme of the closest point detection is based on the seeking for a minimum of the distance function, but on the one hand this minimum does not always exist, and on the other hand there may be several or infinitely many equivalent minimum points. Second, the detection has to be organized in a smart way. Large contact problems imply a large number of contacting nodes on both surfaces, that is why a simple detection technique, based on a comparison of distances from each slave node to all components of the master surface, leads to an excessively time-consuming algorithm, especially if contact elements must be frequently updated.
Segment-to-segment discretization requires totally different detection algorithms based on surface topologies. Since we confine ourself to consideration of the Node-to-Segment contact discretization, the questions of detection for other discretizations will be omitted. The geometrical questions of the closest point definition will be discussed in Chapter2and the detection algorithms will be presented in Chapter3.
1.2.2 Contact discretization
As already mentioned, the contact discretization predetermines the structure of contact elements transferring efforts from one contacting surface to another. Three main types of discretizations may be distinguished:
• Node-to-Node, NTN
• Node-to-Segment, NTS
• Segment-to-Segment, STS
The simplest and the oldest Node-to-Node discretization [Francavilla 75] (Fig.1.2) does not allow any finite sliding or large deformations and introduces restrictions on mesh generation. On the other hand it passes the contact patch test – uniform pressure is transferred correctly through the conforming contact interface. The NTN discretization is applicable for linear and quadratic elements in two dimensional case and only to linear elements in three dimensional case. The NTN technique smoothes the asymmetry between contacting surfaces. However, the normal vector for each pair of nodes is usually determined according to one of the surfaces. Different possibilities of normal definition are presented in Remark3.2in Section3.5.
Figure 1.2: Graphical representation of the Node-to-Node discretization, associated pairs of nodes and corresponding normals constructed on the master surface.
Node-to-segment (Fig. 1.3) is a multipurpose discretization tech-nique [Hughes 77], valid for non-conforming meshes, large deformation and large sliding. But this discretization is not stable and does not pass the contact patch test proposed by Taylor and Papadopoulos [Taylor 91] for non-conforming meshes – a uniform contact pressure cannot be transferred correctly through the contact interface (see. Fig. 1.4). However, this discretization technique passes this patch test in “double pass” for LMM, which means that at each solution step the problem is solved twice: on the first step one assignment of master and slave surfaces is assumed and on the second step the master and slave surfaces are exchanged. A comprehensive discussion of
the contact patch test for NTS discretizations can be found in [Crisfield 00c], where a new approach combining linear and quadratic shape functions is also suggested. Recently, a modification of the NTS discretization has been proposed [Zavarise 09a], which passes the patch test if the PM is used. Besides the drawbacks of this discretization, it is quite simple and robust, that is why it is often implemented in commercial Finite Element Analysis packages. Contact detection and resolution techniques presented in this dissertation are suitable for the NTS discretization.
Figure 1.3: Graphical representation of the Node-to-Segment discretization for different choices of master and slave.
Figure 1.4: a – Scheme of the Taylor contact patch test; b – the resulting nonuniform distribution of the stress component σyy in case of the NTS discretization.
Recently, another technique based on a symmetric Node-to-Segment discretization, the Contact Domain Method, has been proposed in [Oliver 09], [Hartmann 09]. The discretization of the contact interface is based on a full triangulation of the zone between contacting surfaces based on surface nodes (Fig. 1.5). This formulation seems to be rather stable and passes the patch test, but its three dimensional implementation reported in [Oliver 10] is not applicable for arbitrary discretizations of the contacting surfaces.
Figure 1.5: Graphical representation of the discretization in the Contact Domain Method, including a triangulation of the contact interface.
Segment-to-Segmentdiscretization (Fig.1.6) has been first proposed by Simo et al. [Simo 85] for the two dimensional case (see also [Zavarise 98]). Recently such a discretization has been efficiently applied to two and three dimensional problems coupled with the mortar method for nonconforming meshes, inspired by the domain decomposition methods [Wohlmuth 01]. This technique is stable and passes the patch test but its implementation for a general case presents a great challenge, “a nightmare”, according to Tod A. Laursen, one of the authors of the mortar method’s implementation for two and three dimensional both structural and contact problems [Puso 03], [Puso 04], [Yang 05], [Yang 08b], [McDevitt 00].
Figure 1.6: Graphical representation of the Segment-to-Segment discretization, contact elements and an intermediate surface.
A standalone discretization technique is needed for Nitsche method [Becker 03], [Wriggers 08], Gauss points of one surface play the role of slave nodes. The comparison of Nitsche and mortar techniques can be found in [Fritz 04].
The basic idea of the mortar method appeared in the second half of the 80s and in the beginning of 90s for domain decomposition techniques between non-conforming subdomains, see, for example, [Bernardi 90]. In 1998 Belgacem [Belgacem 98] has adapted the mortar method for the multibody or bilateral frictionless contact problem. Further in the beginning of 2000s the rigorous formulation adapted to frictional contact problem subjected to
large deformations and large slidings has been established by McDevitt and
Laursen[McDevitt 00]. The mortar method consists either in introducing an
intermediate contact surface where contact pressure is defined or in using as mortar surface one of the contacting surfaces, for details see [Wriggers 06]. The mortar based formulation leads to a consistent formulation of the frictional contact problem for large sliding and large deformations. It allows to pass the contact patch test for nonconforming meshes and does not suffer from spurious penetrations like the NTS (see Fig.1.7from [Zavarise 98]).
Figure 1.7: Example of spurious penetrations of NTS discretization and accurate treatment in the framework of Segment-to-Segment (adapted from Zavarise and Wriggers [Zavarise 98]).
1.2.3 Contact resolution
As mentioned above, the rigorous formulation of a variational principle for contact problems, results in a variational inequality subjected to geometrical constraints [Kikuchi 88]. These constraints can be brought as additional terms in the objective energy functional by means of penalty, Lagrange multiplier or other methods [Bertsekas 84], [Bertsekas 03], [Luenberger 03], [Bonnans 06], etc. Such an operation converts the constrained optimization, where constraints are given as inequalities, into an unconstrained or partly unconstrained one. If one supposes the active contact zone to be known, then the variational inequality can be replaced by a variational equality, which finally results in an unconstrained problem written in a standard form of variational equality [Wriggers 06]. This problem can be treated as a standard nonlinear minimization problem by means available in the given finite element code. A solver for systems of linear equations and a method for the treatment of nonlinear problems are needed. Note that since the contact constraints are given as inequalities, a special attention has to be paid to the definition of the active contact zone. For penalty and augmented Lagrangian methods, this
task is trivial. For the Lagrange multiplier method, an active set strategy should be employed (for the definition of the active set strategy see, for example [Luenberger 03]).
The resulting unconstrained minimization problem is not sufficiently smooth, which may result in slow convergence of the employed iteration scheme or even in divergence. The stability of the numerical scheme depends on the discretization and on the solution parameters. Note that the frictional contact renders the tangent matrix nonsymmetric, which presents a problem for several solvers (like conjugate gradient method) and for the parallel treatment of the problem: the Schwarz theory for nonsymmetric problems is less satisfactory than for positive definite symmetric problems [Toselli 05]. The way out has been proposed in [Laursen 92], [Laursen 93] for augmented Lagrangian method with Uzawa’s algorithm – governing equations of Coulomb’s friction have been linearized by the operator splitting technique, first recognized in [Glowinski 89], i.e. the entire problem is recast in two subproblems, which are solved once at each solution step. Augmented Lagrangian and Lagrange multiplier methods derive a non-symmetric tangent matrices only for the slip state. This is due to the nonassociativity of Coulomb’s friction law, i.e. slip occurs in the plane of the constant contact pressure. The penalty method suffers from a non-symmetry both in stick and slip states. A solution has been proposed in [Wriggers 06], it consists in a similar treatment of all the deviations from the stick state, i.e. no difference between normal and tangential deviations from the stick are made. Another approach yielding a symmetric tangent matrix in stick state has been proposed in [Konyukhov 05], based on a rigorous covariant description of the contact geometry. The same authors proposed a symmetrization of different friction models based on the augmented Lagrangian method coupled with Uzawa’s algorithm [Konyukhov 07b].
1.3
Motivation and overview
The principal motivation of this dissertation is implementation of a robust and fast sequential and parallel contact algorithm in the implicit Finite Element software Z-set (ZéBuLoN) [Besson 97]. Since there is no specific predefined application, the algorithm should be multipurpose. The principle requirement is the efficient treatment of large contact problems within sequential or parallel framework on parallel computers with distributed memory and within the nonoverlapping domain decomposition methods. Another aim predefined the size of the manuscript is to provide a reader with a consistent theoretical and methodological foundation of the computational contact mechanics.
The three principal parts of the thesis are: geometry of the contact, contact detection and resolution techniques. All parts are interdependent, but we tried to render each chapter more or less self-sufficient. So the sequence order of chapters is rather arbitrary. It is worth mentioning that in all chapters, except the chapter devoted to contact detection, a new algebra is used. It arises from a generalization of the tensor algebra and operates with abstract s-structures, elements of s-spaces. A comprehensive presentation of s-structures in a general
context is given in Appendix A. It follows the presentation of vectors and higher-order tensors which is given in a form conforming with s-structures.
Chapter 2 gives an elaborated presentation of the geometrically precise theory of contact. The relevant geometrical quantities and notions are introduced: closest points, gap function, tangential relative sliding. Several original ideas related to the geometrically precise theory of contact are presented.
In Chapter 3, different contact detection schemes for Node-to-Segment discretization are elaborated in minute detail. Sequential and parallel implementation are discussed both for known and unknown a priori master-slave discretizations. Some numerical examples are given.
The main governing equations are given in Chapter4. The presentation starts from the primitive case of a unilateral contact with a rigid plane. Based on this simple case an interpretation of frictional and contact constraints is given by ordinary Dirichlet and Neumann boundary conditions. Next, Signorini’s problem is presented and finally a bilateral (multibody) framework is given. The resulting variational inequality is recast in a variational equality using penalty, Lagrange multiplier and augmented Lagrangian methods. Related weak forms for frictionless and frictional contact for each method are presented and the resulting algorithm is illustrated by a simple example of unilateral contact.
Chapter5provides the reader with a minimal knowledge of the numerical schemes used in computational contact mechanics. Newton’s method and its generalization for the case of nonsmooth functions and the return mapping algorithm are presented. Next, a short introduction in the standard formalism of the Finite Element Method is given, followed by the derivation of the expressions needed for implementing of the penalty and augmented Lagrangian method in a finite element code. Linearized forms adapted for the Newton’s method are deduced. Finally, the details of implementation of the partial Dirichlet-Neumann approach considered in Chapter4 are discussed, followed by several technical remarks on the implementation of the contact algorithms in a Finite Element environment.
Numerical examples are brought together in Chapter6. Contact problems with known analytical solution and examples demonstrating the performance of the implemented methods for highly nonlinear problems are presented. Finally, in Chapter7the main contributions and the short term perspectives of the dissertation are summarized.
Geometry in contact mechanics
Résumé de Chapitre
2
«Géométrie en mécanique de
contact»
Le deuxième chapitre présente un travail fondamental et original portant sur la définition d’une théorie précise de la géométrie du contact. Après avoir démontré l’importance de la description géométrique et les ambiguïtés qui peuvent être liées à des situations pathologiques, on introduit quelques définitions de base telles que la distance de séparation, le point le plus proche et la vitesse tangentielle relative. On discute en détail les subtilités (asymétrie, non-unicité, existence) de la définition du point le plus proche et la différence entre le minimum et infimum dans cette définition. Puis on donne la forme qui définit de façon rigoureuse la distance de séparation normale ; cette forme est adaptée aux surfaces lisses par morceaux et est bien adaptée aux méthodes numériques de détection.
Cependant la séparation normale et la méthode de projection associée ont quelques inconvénients : en particulier, le point de projection n’est pas une fonction continue de la position du point esclave. La distance de glissement est associée directement à deux positions consécutives de la projection et l’énergie dissipée due au frottement est liée à la distance de glissement. Ainsi la forme faible du système devient-elle non continue, si bien que la convergence de la résolution n’est pas assurée. Ceci est un argument pour construire une nouvelle procédure de projection, dite «ombre portée». Le point de projection de l’esclave sur la surface maître est construit en fonction de l’ombre projetée du point esclave, la lumière étant émise par un point imaginaire, situé à une distance finie, ou à l’infini. Cette procédure permet de retrouver une projection continue et, en conséquence une forme faible mieux adaptée à la résolution numérique.
Puis on dérive les première et deuxième variations des variables cinétiques (la séparation normale et le paramètre local) pour la géométrie continue. Tous les calculs ont été faits en exploitant un nouveau formalisme, dit «algèbre de S-structure», qui est une généralisation des opérations sur les tenseurs à des opérations sur des champs des tenseurs. On obtient toutes les expressions nécessaires pour intégrer une géométrie arbitraire et non-linéaire dans un code de calcul par élément finis pour la projection normale et la projection d’ombre portée. Les expressions complètes sont comparées d’une part avec les «vraies» variations obtenues numériquement, et avec les expressions analytiques simplifiées souvent utilisées dans le calcul numérique. L’étude statistique démontre l’avantage et la convergence des formes rigoureuses.
En fin de ce chapitre, on propose une nouvelle méthode d’enrichissement de la géométrie de contact, inspirée par la méthode X-FEM. Le but est de prendre en compte une géométrie complexe de la surface sans la discrétiser. Cette méthode est utile pour la simulation du frottement anisotrope et dans le cas où la géométrie change localement en raison d’un changement d’état de déformation et de contrainte (usure, mécanismes d’intrusion–extrusion, etc. . . ).
2.1
Introduction
Contact phenomenon takes place at the interface between solids. This fact implies a strong connection of the contact problem with a rigorous description of the geometry of contacting surfaces. The first continuum based description of the contact problem was given by Simo and Laursen [Laursen 93] and Laursen [Laursen 94]. Such a geometrical description still presents an interesting topic for research in computational contact mechanics, see e.g.recent articles by Konyukhov [Konyukhov 06b], [Konyukhov 06a], [Konyukhov 09]. The mathematical formulation of frictionless contact conditions leads to equations connecting the normal contact pressure σnwith the mutual penetration of bodies, expressed by a signed gap function, frequently the normal gap function, gnis used. The formulation of frictional contact leads to the connection between shear or tangential contact stress vectorσtand the relative tangential sliding velocity
˙gt. The contact stress has to be integrated over the contact surface Γci of each solid, where i ∈ [0; Nc] and Ncthe total number of contacting surfaces.
Let us show how important the geometry is for contact mechanics and how the geometrical description may predetermine methods and approaches which are used in numerical treatment of contact. As is known, there is an ambiguity in the definition of the normal gap gnbetween contacting surfaces. At first glance, it seems easy to determine the normal gap for each point of a contacting surface as a distance to the closest point of the second surface: for a point M of the first surface M ∈ ∂A one seeks for the closest point N on the other surface N ∈ ∂B. Three problems arise from such a definition:
P1 Asymmetry of surfaces (Fig.2.1,a.)
If instead of seeking for the closest point N ∈ ∂B to the point M ∈ ∂A, one inverts the problem and searches for the point M′ ∈ ∂A closest to
the N ∈ ∂B, then the points M and M′ do not coincide as soon as the contacting surfaces are not parallel at least locally. It means that there is no one-to-one equitable correspondence between surface points. It implies an asymmetry in the gap function and consequently in the entire geometrical description.
P2 Non-uniqueness of the closest point (Fig.2.1,b.)
For example, the center of a circle does not have a single closest point on the circle, but all the points on the circle are equally close to its center. All other points have a unique closest point on the circle. The uniqueness of the closest point refers to the curvature of the considered curve or surface and has been discussed in detail by Heergaard and Curnier [Heegaard 96], Pietrzak [Pietrzak 97] and Konyukhov [Konyukhov 08]. The limit case of the infinite curvature corresponds to the third problem.
P3 Requirement of smoothness (Fig.2.1,c.)
Smoothness of at least one of the contacting surfaces (master) is not sufficient but necessary condition for existence of the normal projection1 point. The smooth surface allows a rigorous mathematical description of contact, a robust detection procedure and a reliable convergence of numerical schemes. However, surfaces in the Finite Element framework are only piecewise smooth due to the discretization. The non-smoothness represents another source for existence of multiple closest points and generates blind angles in normal projection domains, discontinuous normal vector field and related problems - oscillations and possible divergence of the numerical solution.
Figure 2.1: Geometry related problems: a – asymmetry of the closest point definition; b – non-uniqueness of the closest point; c – nonexistence of the normal projection point.
All these difficulties affect the geometrical description of the contact. The asymmetry of the closest point detection (P1) results in an asymmetric
1by normal projection of a slave we imply such a point on the master surface that the vector
connecting the slave and its projection is collinear with the normal vector constructed on the master surface
treatment of contact surfaces, it leads to the so-called “master-slave” approach (also called “target-impactor” or “target-contactor”). For each point of the slave surface ri ∈ Γs the closest point on the master surface ρ
i ∈ Γm has to be determined, i.e. ρ i= ρ i∈ Γm ∀ρ ∈ Γm : kρ i− rik ≤ kρ − rik .
Due to the non-uniqueness of the closest point (P2), a different technique fulfilling additional conditions on the uniqueness and continuity of the projection, can be elaborated (see section 2.2.3). That allows to improve the convergence and to avoid nonphysical discontinuities in sliding path, which is crucial for a rigorous description of frictional contact.
The non-smoothness of the surface (P3) arises from the discontinuity of local bases across boundaries of adjacent segments or faces in the finite element discretization and produces convergence problems and oscillations in the finite element framework. The main remedy consists in smoothing the master surface over several segments [Pietrzak 97], [Wriggers 01], [Krstulovi´c-Opara 02].
As one can see, the definition of the gap function and the closest point brings out a series of difficulties. This short preface allows to realize the importance of a well founded geometrical approach needed to deal with contact problems. Especially, it worth mentioning that classical contact detection techniques are strongly connected with the closest point definition. All results concerning the definition of the closest point will be used in Chapter 3 devoted to the development of a reliable contact detection procedure.
The aim of this chapter is an elaborated analysis of geometry related questions in computational contact mechanics in the framework of the FEM and the Node-to-Segment discretization. According to the high standards which were set up in computational contact mechanics by Simo and Laursen [Laursen 93] and in order to provide a multipurpose and discretization independent framework, we start from the continuous description of the contact geometry. Such an approach is valid both for classical Node-to-Segment discretization for any type of finite elements and for special cases, namely unilateral contact with a rigid surface and smoothed master surfaces:
• Unilateral contact with a rigid surface:
contact between a deformable solid and a rigid surface, the latter can be described by an analytical function or a CAD model, see [Hansson 90] for frictionless and [Heege 96] for frictional cases; among engineering problems subjected to this case there are metal forming and metal processing, rubber-metal and tire-road contact, etc.
• Smoothed master surface: for many reasons it is advantageous to replace a piecewise smooth master surface by a C1 smooth surface (NURBS, Bézier, Gregory patches, etc) based on information from several adjacent master segments; this procedure ensures a continuous projection on the master surface and leads to a better convergence [Pietrzak 97], [Padmanabhan 01], [Puso 02], [Wriggers 01], [Krstulovi´c-Opara 02] and
![Figure 1.7: Example of spurious penetrations of NTS discretization and accurate treatment in the framework of Segment-to-Segment (adapted from Zavarise and Wriggers [ Zavarise 98 ]).](https://thumb-eu.123doks.com/thumbv2/123doknet/2992358.83685/36.892.158.716.338.601/example-penetrations-discretization-treatment-framework-zavarise-wriggers-zavarise.webp)
